| import OAI.Probability.DilutedSpin.CavityRateLimits |
| import OAI.Probability.DilutedSpin.FullRegularSelection |
| import OAI.Probability.DilutedSpin.ScoreIncrement |
|
|
| namespace OAI |
|
|
| section |
| namespace DilutedSpinGlass |
| open Filter |
| open scoped Topology |
|
|
| lemma scoreScale_div_succ_tendsto : |
| Tendsto (fun N => scoreScale N/(N+1)) atTop (𝓝 0) := by |
| have he := (tendsto_natCast_atTop_atTop : Tendsto (fun N : ℕ => (N:ℝ)) atTop atTop) |
| have ht := (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ)<1/4)).comp he |
| apply squeeze_zero' (Eventually.of_forall (fun N => by unfold scoreScale; positivity)) |
| (eventually_atTop.2 ⟨1,fun N hN => ?_⟩) ht |
| change scoreScale N/(N+1) ≤ (N:ℝ)^(-(1/4:ℝ)) |
| rw [← neg_div,← SizeCoupling.scoreScale_div (show 0 < N by omega)] |
| apply div_le_div_of_nonneg_left |
| · unfold scoreScale; positivity |
| · exact_mod_cast (show 0 < N by omega) |
| · linarith |
|
|
| namespace UniversalDictionary |
| lemma gridExponents_exponents (L : ℕ) : |
| Exponents (fun i : Fin L => gridExponents L i.castSucc) := by |
| constructor |
| · intro i j hij |
| simp only [gridExponents,Fin.val_castSucc,Nat.cast_add,Nat.cast_one] |
| apply (div_lt_div_iff_of_pos_right (by positivity : (0:ℝ)<L+1)).mpr |
| exact_mod_cast (show i.val+1<j.val+1 by exact Nat.add_lt_add_right hij 1) |
| · intro i |
| refine ⟨gridExponents_pos L i.castSucc,?_⟩ |
| simp only [gridExponents,Fin.val_castSucc,Nat.cast_add,Nat.cast_one] |
| apply (div_lt_one (by positivity : (0:ℝ)<L+1)).mpr |
| exact_mod_cast (show i.val+1<L+1 by omega) |
| end UniversalDictionary |
| end DilutedSpinGlass |
|
|
| end |
|
|
| end OAI |
|
|
